## The alldifferent Constraint: A Survey (2001)

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### BibTeX

@MISC{Hoeve01thealldifferent,

author = {W. J. van Hoeve},

title = {The alldifferent Constraint: A Survey},

year = {2001}

}

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### Abstract

The constraint of difference is known to the constraint programming community since Lauriere introduced Alice in 1978. Since then, several strategies have been designed to solve the alldifferent constraint. This paper surveys the most important developments over the years regarding the alldifferent constraint. First we summarize the underlying concepts and results from graph theory and integer programming. Then we give an overview and an abstract comparison of different solution strategies. In addition, the symmetric alldifferent constraint is treated. Finally, we show how to apply cost-based filtering to the alldifferent constraint.

### Citations

1600 |
Theory of Linear and Integer Programming
- Schrijver
- 1986
(Show Context)
Citation Context ...eorem 18, the soft alldifferent(x1,...,xn,z,µdec) is hyper-arc consistent. 6. The Alldifferent Polytope One of the cornerstones of Operations Research is the field of integer linear programming; see (=-=Schrijver, 1986-=-) and (Nemhauser & Wolsey, 1988). An integer linear programming model consists of integer variables, a set of linear constraints (inequalities or equations) and a linear objective function to be optim... |

1185 |
L.: Integer and Combinatorial Optimization
- Nemhauser, Wolsey
- 1988
(Show Context)
Citation Context ...different(x1,...,xn,z,µdec) is hyper-arc consistent. 6. The Alldifferent Polytope One of the cornerstones of Operations Research is the field of integer linear programming; see (Schrijver, 1986) and (=-=Nemhauser & Wolsey, 1988-=-). An integer linear programming model consists of integer variables, a set of linear constraints (inequalities or equations) and a linear objective function to be optimized. An integer linear program... |

1126 | Depth search and linear graph algorithms
- Tarjan
- 1972
(Show Context)
Citation Context ... have to search for the so-called strongly connected components of the graph [18]. For this problem we can use an implementation by Tarjan that runs in O(n +m) time on graphs with n nodes and m edges =-=[18, 21]-=-. In the algorithm from Figure 4, the search for a maximum matching remains the dominant factor, hence the total algorithm runs in O( # |XC |m) time. The notion of hyper-arc consistency was introduced... |

865 | The Hungarian method for the assignment problem
- Kuhn
- 1955
(Show Context)
Citation Context ...ferent constraint and the objective function as a minimum assignment problem and ignoring the other constraints. The minimum assignment problem can for instance be solved with the Hungarian Algorithm =-=[18, 4]-=-. An important feature article.tex; 22/11/2001; 15:10; p.21 22 of the Hungarian Algorithm is that it is incremental. The first time it is called it runs in O(n 3 ) time and all next computations run i... |

631 |
Combinatorial Optimization: Polyhedra and Efficiency. Volume 24 of Algorithms and Combinatorics
- Schrijver
- 2003
(Show Context)
Citation Context ...e results are well-known and there already exist excellent overviews of these topics. More information about matching theory can for example be found in (Lovász & Plummer, 1986), (Gerards, 1995) and (=-=Schrijver, 2003-=-, Chapter 16–38). More information about flow theory can be found in (Ahuja, Magnanti, & Orlin, 1993) and (Schrijver, 2003, Chapter 6–15). However, we do present proofs and algorithms that provide use... |

621 |
Network Flows
- Ahuja, Magnanti, et al.
- 1993
(Show Context)
Citation Context ... More information about matching theory can for example be found in (Lovász & Plummer, 1986), (Gerards, 1995) and (Schrijver, 2003, Chapter 16–38). More information about flow theory can be found in (=-=Ahuja, Magnanti, & Orlin, 1993-=-) and (Schrijver, 2003, Chapter 6–15). However, we do present proofs and algorithms that provide useful insights. In this paper, we mainly follow the notation of (Schrijver, 2003), unless this conflic... |

620 |
Constraint processing
- Dechter
- 2003
(Show Context)
Citation Context ... . . . . . 33 6 The Alldifferent Polytope 35 References 37 21. Introduction Many combinatorial (optimization) problems can be modeled and solved using the constraint programming paradigm (Apt, 2003; =-=Dechter, 2003-=-). In constraint programming, a model is stated by means of variables that range over their domain of possible values, and constraints on these variables. A constraint restricts the space of possible ... |

592 |
An n(5/2) algorithm for maximum matching in bipartite graphs
- Hopcroft, Karp
- 1975
(Show Context)
Citation Context ...h GV . This can be done for instance with a so-called augmenting path algorithm. Hopcroft and Karp gave an implementation for this that runs in O( # |XC |m) time, where m is the number of edges of GV =-=[16]-=-. Their algorithm still remains essentially the best known [6]. From Hall's Theorem we already know that whenever we find a subset of nodes the cardinality of which exceeds the cardinality of the corr... |

569 |
Matching Theory
- Lovasz, Plummer
- 1986
(Show Context)
Citation Context ...roblem in graph theory. We will first give an illustrative example from which the alldifferent constraint can be easily expressed as a maximum matching problem. Good references to matching theory are =-=[21, 12]-=-. EXAMPLE 3 (Task assignment). In this small example we want to assign four tasks to five machines. To each machine at most one task can be assigned. However, not every task can be assigned to every m... |

551 |
Graphs and hypergraphs
- Berge
- 1973
(Show Context)
Citation Context ...text. Note the use of braces ({, }) and brackets ([, ]) that indicate a set and an interval of integer domain values respectively. Thus, the set {1, 3} contains the vales 1 and 3 whereas the interval =-=[1, 3]-=- contains 1, 2 and 3. DEFINITION 1 (Arc consistency). A binary constraint C(x 1 , x 2 ) where D 1 and D 2 are non-empty, is called arc consistent i# #d 1 # D 1 #d 2 # D 2 such that (d 1 , d 2 ) # C, a... |

408 |
Constraint Satisfaction in Logic Programming. Logic ProgrammingSeries
- Hentenryck
- 1989
(Show Context)
Citation Context ...l overview and an abstract comparison of these di#erent strategies. 1 Introduction Many problems from combinatorial optimization can be modeled and solved using techniques from Constraint Programming =-=[14, 22]-=-. One of the constraints that arises naturally in these models is the alldifferent constraint, which states that all variables in this constraint must be pairwise di#erent. In Example 1, a scheduling ... |

389 |
Programming with Constraints : An Introduction
- Marriott, Stuckey
- 1998
(Show Context)
Citation Context ...hers. Printed in the Netherlands. article.tex; 22/11/2001; 15:10; p.1 2 1. Introduction Many combinatorial optimization problems can be modeled and solved using techniques from constraint programming =-=[22, 33]-=-. One of the constraints that arises naturally in these models is the alldifferent constraint, which states that all variables in this constraint must be pairwise di#erent. In this section we first in... |

354 |
On representatives of subsets
- Hall
- 1935
(Show Context)
Citation Context ..., #t # T # t#T a tm y tm # 1, #m # M y tm # {0, 1} #t # T , #m # M. 3.3. Hall's theorem A useful theorem to derive algorithms that ensure consistency for the alldifferent constraint is Hall's Theorem =-=[13]-=-. The following formulation is stated in terms of the alldifferent constraint. The cardinality of a set K is denoted by |K|. THEOREM 1 (Hall). The constraint alldifferent(x 1 , . . . , x n ) with resp... |

346 |
A filtering algorithm for constraints of difference in CSPs
- Régin
- 1994
(Show Context)
Citation Context ... sequence of not-equal constraints; see for example (Van 4Hentenryck, 1989). Unfortunately the global information is lost in that way. The global view was retrieved with the algorithm introduced by (=-=Régin, 1994-=-), that considers all notequal constraints simultaneously. Throughout the history of constraint programming, the alldifferent constraint has played a special role. Various papers and books make use of... |

287 | Limited discrepancy search
- Harvey, Ginsberg
- 1995
(Show Context)
Citation Context ...junction with an objective function has been done in (Lodi, Milano, & Rousseau, 2003). In that work, the so-called additive bounding procedure (Fischetti & Toth, 1989) and limited discrepancy search (=-=Harvey & Ginsberg, 1995-=-) are exploited in presence of an alldifferent constraint. 5.3 The Soft Alldifferent Constraint Consider a CSP that is over-constrained, i.e. there exists no solution satisfying all constraints. In su... |

264 | The constraint logic programming language CHIP - Dincbas, Hentenryck, et al. - 1988 |

247 | Combinatorial optimization
- Cook, Cunningham, et al.
- 1998
(Show Context)
Citation Context ...g path algorithm. Hopcroft and Karp gave an implementation for this that runs in O( # |XC |m) time, where m is the number of edges of GV [16]. Their algorithm still remains essentially the best known =-=[6]-=-. From Hall's Theorem we already know that whenever we find a subset of nodes the cardinality of which exceeds the cardinality of the corresponding set of domain values, no matching exists that satura... |

207 | Principles of Constraint Programming
- Apt
(Show Context)
Citation Context .... . . . . . . . . . . 33 6 The Alldifferent Polytope 35 References 37 21. Introduction Many combinatorial (optimization) problems can be modeled and solved using the constraint programming paradigm (=-=Apt, 2003-=-; Dechter, 2003). In constraint programming, a model is stated by means of variables that range over their domain of possible values, and constraints on these variables. A constraint restricts the spa... |

183 | Generalized arc-consistency for global cardinality constraint
- Régin
- 1996
(Show Context)
Citation Context ...alldifferent constraint to be partially violated. 5.1 The Symmetric Alldifferent Constraint A particular case of the alldifferent constraint, the symmetric alldifferent constraint, was introduced by (=-=Régin, 1999-=-b). We assume that the variables and their domain values represent the same set of elements. The symmetric alldifferent constraint states that all variables must take different values, and if the vari... |

152 |
An O( √|V ||E|) algorithm for finding maximum matching in general graphs
- MICALI, VAZIRANI
- 1980
(Show Context)
Citation Context ...me complexity of O(|V | 1/2 |A|). In a general graph G = (V,E) (not necessarily bipartite), a maximum-size matching can be computed in O(|V | |E|) time (Edmonds, 1965) 4 or even O(|V | 1/2 |E|) time (=-=Micali & Vazirani, 1980-=-). 2.2.3 Flow Theory Let G = (V,A) be a directed graph. For v ∈ V , let δ in (v) and δ out (v) denote the multiset of arcs entering and leaving v, respectively. Let s,t ∈ V denote the “source” and the... |

115 |
Good old discrete relaxation
- Mohr, Masini
- 1988
(Show Context)
Citation Context ...rom Figure 4, the search for a maximum matching remains the dominant factor, hence the total algorithm runs in O( # |XC |m) time. The notion of hyper-arc consistency was introduced by Mohr and Masini =-=[16]-=-. They also give a general algorithm to achieve this notion. For an n-ary alldifferent constraint, where the domain size of all variables is bounded by d, D i # d, the time complexity of the general a... |

113 | ECLiPSe: A platform for constraint logic programming
- Wallace, Novello, et al.
- 1997
(Show Context)
Citation Context ...p (Dincbas, Van Hentenryck, Simonis, Aggoun, Graf, & Berthier, 1988), it was also possible to express the constraint of difference as the well-known alldifferent constraint. In the system Ecl i ps e (=-=Wallace, Novello, & Schimpf, 1997-=-) this constraint was introduced as alldistinct. However, in the early constraint (logic) programming systems this constraint was treated internally as a sequence of not-equal constraints; see for exa... |

112 | flowers
- Edmonds, ”Paths
- 1965
(Show Context)
Citation Context ...wever, we can article.tex; 22/11/2001; 15:10; p.20 21 apply similar reasoning to this case. There are algorithms for finding a maximum matching in nonbipartite graphs, for instance the one by Edmonds =-=[9]-=-. Regin proposes an algorithm that makes a symm alldifferentsconstraint hyper-arc consistent that has a running time of O(nm), together with an algorithm that has a time complexity of O(m) but does no... |

98 | The essence of constraint propagation
- APT
- 1999
(Show Context)
Citation Context ...we do not obtain a globally consistent CSP, but a CSP in which all constraints are locally, i.e. individually, consistent. A thorough description of the process of constraint propagation is given in (=-=Apt, 1999-=-); see also (Apt, 2003). After splitting a CSP, domain filtering and constraint propagation is applied to the smaller CSPs. The removal of domain values leads to a smaller search tree and thus speeds ... |

93 |
Two theorems in graph theory
- Berge
- 1957
(Show Context)
Citation Context ...nsider the graph G ′ = (V,M ∪ M ′ ). In G ′ , each vertex is connected to at most two edges. Hence, each component of G ′ is either a circuit or 3. In the literature this result is often ascribed to (=-=Berge, 1957-=-). However, it should actually be attributed to Petersen, as for example pointed out by (Mulder, 1992). 10a path (possibly of length zero). As |M ′| > |M| there is at least one component containing m... |

92 | On the equivalence of constraint satisfaction problems
- Rossi, Petrie, et al.
- 1990
(Show Context)
Citation Context ...ng new variables on which we define (some of) the binary constraints in Cdec. In that case we apply a mapping of the solution set of Tk i=1 Ci to the solution set of C and vice versa, as proposed by (=-=Rossi, Petrie, & Dhar, 1990-=-). In this paper this extension is not necessary, however. 16Theorem 6 Algorithm 1 establishes arc consistency on the binary decomposition of alldifferent(x1,x2,... ,xn) or proves that it is inconsis... |

88 |
A language and a program for stating and solving combinatorial problems
- LAURIERE
- 1978
(Show Context)
Citation Context ...traints than we can apply to the set of disequality constraints. 1.2. Historical overview In 1978, Lauriere introduced Alice, "A language and a program for stating and solving combinatorial probl=-=ems" [19]. Already -=-in this system the importance of the alldifferent constraint was recognized. The keyword "DIS" applied to a set of variables is used to state that the variables must take di#erent values. It... |

76 | Global constraints as graph properties on a structured network of elementary constraints of the same type
- Beldiceanu
- 2000
(Show Context)
Citation Context ...rogramming are used to achieve some kind of consistency [17, 24]. Over the years, the alldifferent constraint as well as other global constraints were well-studied in constraint programming (see e.g. =-=[2]-=- for an overview). Special algorithms were being developed that are able to exploit the global information of the constraints. For the alldifferent constraint at least five di#erent filtering algorith... |

76 |
Logic-Based Methods for Optimization: Combining Optimization and Constraint Satisfaction
- Hooker
(Show Context)
Citation Context ...gorithms exist, each achieving a di#erent kind of consistency, or achieving it faster [28, 20, 26, 23]. Although the constraint is mentioned more or less deeply in a variety of papers and books (e.g. =-=[27, 15, 22]-=-), to our knowledge there is no work that collects all e#ort that has been put into this constraint. This paper therefore tries to give an overview of the alldifferent constraint, which will be outlin... |

72 | Introducing global constraints - Beldiceanu, Contejean - 1994 |

65 |
A fast algorithm for the bound consistency of alldi constraints
- Puget
- 1998
(Show Context)
Citation Context ...ise di#erent. In Example 1, a scheduling problem is modeled using the alldifferent constraint. Example 1 (Scheduling of speeches). Consider the following simple scheduling problem, adapted from Puget =-=[17]-=-, where a set of speeches must be scheduled during one day. Each speech lasts exactly one hour (including questions and a co#ee break), and only one conference room is available. Furthermore, each spe... |

64 | Local and global relational consistency
- Dechter, Beek
- 1997
(Show Context)
Citation Context ... set of disequalities that is equivalent to the hyper-arc consistency notion for the alldifferent constraint. Relational consistency can be used for this. DEFINITION 9 (Relational (1, m) consistency, =-=[8]-=-). A set of constraints S = {C 1 , . . . , Cm } is relationally (1, m)-consistent i# all domain values d # D i of variables appearing in S, appear in a solution to the m constraints, evaluated simulta... |

57 |
Die Theorie der regulären graphs
- Petersen
(Show Context)
Citation Context ...ating if its edges are alternatingly out of and in M. On an M-augmenting path, we can exchange edges in M and not in M, to obtain a matching M ′ with |M ′ | = |M| + 1. The following result is due to (=-=Petersen, 1891-=-) 3 . Theorem 2 Let G = (V,E) be a graph, and let M be a matching in M. Then either M is a maximum-size matching, or there exists an M-augmenting path. Proof. If M is a maximum-size matching, then the... |

48 |
Cost-based domain filtering
- Focacci, Lodi, et al.
- 1999
(Show Context)
Citation Context ...d exceeds the upper bound, we can abandon the current branch and the underlying subtree, because this subtree cannot contain a better solution than the currently best one. Cost-based domain filtering =-=[10]-=- makes use of this property of branchand -bound, but applies it as a domain filtering technique from constraint programming. Consider a CSP that contains an alldifferent constraint, together with an o... |

48 |
Global constraint catalog
- Beldiceanu, Carlsson, et al.
- 2005
(Show Context)
Citation Context ... and the global cardinality constraint (Régin, 1996). Over the years, the alldifferent constraint as well as other global constraints has been well-studied in constraint programming; see for example (=-=Beldiceanu, Carlsson, & Rampon, 2005-=-) and (Régin, 2003) for an overview. Special algorithms have been developed that are able to exploit the global information of the constraints. As we will see, for the alldifferent constraint at least... |

46 | Specific Filtering Algorithms for Over Constrained Problems
- Petit, Régin, et al.
(Show Context)
Citation Context ...oblem (COP), where all constraints are hard, and the (weighted) sum of cost variables is minimized. This approach allows one to use specialized filtering algorithms for soft constraints, as shown by (=-=Petit, Régin, & Bessière, 2001-=-). We first give a general definition of constraint softening, which we later apply to the alldifferent constraint. 31Definition 15 (Constraint softening) Let x1,x2,... ,xn,z be variables with respec... |

40 | A fast and simple algorithm for bounds consistency of the alldierent constraint - Lopez-Ortiz, Quimper, et al. - 2003 |

37 | Solving various weighted matching problems with constraints
- Caseau, Laburthe
- 1997
(Show Context)
Citation Context ...e problem we want to solve involves also other constraints), the goal is to find a solution with minimum 8 total cost. In the literature, this combination is known as the constraint MinWeightAllDiff (=-=Caseau & Laburthe, 1997-=-), or IlcAllDiffCost (Focacci, Lodi, & Milano, 1999). This section shows how to exploit the alldifferent constraint and the minimization problem together as an “optimization constraint”. First we give... |

36 | Faster Algorithms for Bound-Consistency of the Sortedness and the AllDierent Constraint
- Mehlhorn, Thiel
- 2000
(Show Context)
Citation Context ...oit the global information of the constraints. For the alldifferent constraint at least five di#erent filtering algorithms exist, each achieving a di#erent kind of consistency, or achieving it faster =-=[28, 20, 26, 23]-=-. Although the constraint is mentioned more or less deeply in a variety of papers and books (e.g. [27, 15, 22]), to our knowledge there is no work that collects all e#ort that has been put into this c... |

36 | The difference all-difference makes
- Stergiou, Walsh
(Show Context)
Citation Context ...en the alldifferent constraint is being made hyper-arc consistent, all variables are considered at the same time, which allows a much stronger local consistency. This is shown in Theorem 7; see also (=-=Stergiou & Walsh, 1999-=-). Theorem 7 Let P be a CSP and Pdec the same CSP in which all alldifferent constraints have been replaced by their binary decomposition. Then ΦHA(P) ≼ ΦA(Pdec). Proof. To show that ΦHA(P) ≼ ΦA(Pdec),... |

32 | A scheme for unifying optimization and constraint satisfaction methods
- Ottosson, Thorsteinsson, et al.
- 2000
(Show Context)
Citation Context ...ot of attention recently. A common example to introduce this field uses the alldifferent constraint, where methods of graph theory and integer programming are used to achieve some kind of consistency =-=[17, 24]-=-. Over the years, the alldifferent constraint as well as other global constraints were well-studied in constraint programming (see e.g. [2] for an overview). Special algorithms were being developed th... |

32 |
A bounds-based reduction scheme for constraints of dierence
- Leconte
- 1996
(Show Context)
Citation Context ...oit the global information of the constraints. For the alldifferent constraint at least five di#erent filtering algorithms exist, each achieving a di#erent kind of consistency, or achieving it faster =-=[28, 20, 26, 23]-=-. Although the constraint is mentioned more or less deeply in a variety of papers and books (e.g. [27, 15, 22]), to our knowledge there is no work that collects all e#ort that has been put into this c... |

31 |
Algorithms and Codes for the Assignment Problem
- Carpaneto, Martello, et al.
- 1988
(Show Context)
Citation Context ...ferent constraint and the objective function as a minimum assignment problem and ignoring the other constraints. The minimum assignment problem can for instance be solved with the Hungarian Algorithm =-=[18, 4]-=-. An important feature article.tex; 22/11/2001; 15:10; p.21 22 of the Hungarian Algorithm is that it is incremental. The first time it is called it runs in O(n 3 ) time and all next computations run i... |

30 |
A algorithm for constraints of dierence in CSPs
- Regin
- 1994
(Show Context)
Citation Context ...relatively high, namely around O(n 2 ), whereas the hyper-arc consistency algorithms are around O(dn 1.5 ), where d is the maximum cardinality of the domains and n is the number of variables involved =-=[12, 18]-=-. Nevertheless, this filtering algorithm applies quite well to several problems, such as the n-queens problem (ns200) [12, 17]. Other work on the comparison of the alldifferent constraints and the cor... |

25 |
Completing Quasigroups or Latin Squares: A Structured Graph Coloring Problem
- Gomez, Shmoys
(Show Context)
Citation Context ...r when a directed graph has to be covered with disjoint circuits. Numerous applications exist in which the alldifferent constraint is of vital importance, for example quasi-group completion problems (=-=Gomes & Shmoys, 2002-=-), air traffic management (Barnier & Brisset, 2002; Grönkvist, 2004) and rostering problems (Tsang, Ford, Mills, Bradwell, Williams, & Scott, 2004). Finally, many other global constraints can be viewe... |

24 | Comparing partial consistencies
- Collavizza, Delobe, et al.
- 1999
(Show Context)
Citation Context ... i. When both P # P # and P # # P we write P # P # . A failed CSP, i.e. a CSP with at least one empty domain, is denoted by P # . By convention, P # is the smallest CSP. This notation is adopted from =-=[5]-=-. PROPOSITION 1. #HA (P ) # #R (P ) # #B (P ). Proof. Both hyper-arc consistency and range consistency verify all values of all domains. But hyper-arc consistency verifies the constraints with respect... |

24 |
An additive bounding procedure for combinatorial optimization problems
- Fischetti, Toth
- 1989
(Show Context)
Citation Context ...Other work concerning the alldifferent constraint in conjunction with an objective function has been done in (Lodi, Milano, & Rousseau, 2003). In that work, the so-called additive bounding procedure (=-=Fischetti & Toth, 1989-=-) and limited discrepancy search (Harvey & Ginsberg, 1995) are exploited in presence of an alldifferent constraint. 5.3 The Soft Alldifferent Constraint Consider a CSP that is over-constrained, i.e. t... |

24 | When do bounds and domain propagation lead to the same search space
- Schulte, Stuckey
- 2001
(Show Context)
Citation Context ...roposed by (Lopez-Ortiz et al., 2003) appears to be most efficient in practice. A general comparison of bounds consistency and hyper-arc consistency with respect to the search space has been made by (=-=Schulte & Stuckey, 2001-=-). In particular, attention is also paid to the alldifferent constraint. 5. Variants of the Alldifferent Constraint This section presents three variants of the alldifferent constraint: the symmetric a... |

23 | Completing quasigroups or latin squares: A structured graph coloring problem - Gomes, Shmoys - 2002 |

21 | Graph Coloring for Air Traffic Flow Management
- Barnier, Brisset
- 2004
(Show Context)
Citation Context ...disjoint circuits. Numerous applications exist in which the alldifferent constraint is of vital importance, for example quasi-group completion problems (Gomes & Shmoys, 2002), air traffic management (=-=Barnier & Brisset, 2002-=-; Grönkvist, 2004) and rostering problems (Tsang, Ford, Mills, Bradwell, Williams, & Scott, 2004). Finally, many other global constraints can be viewed as an extension of the alldifferent constraint, ... |